6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
6: Fourier Transform
Fourier Transform: 6 – 1 / 12
Fourier Series as T → ∞
6: Fourier Transform
• Fourier Series as
Fourier Series: u(t) =
T →∞
• Fourier Transform
• Fourier Transform
P∞
n=−∞
Un ei2πnF t
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 2 / 12
Fourier Series as T → ∞
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Fourier Series: u(t) =
P∞
n=−∞
Un ei2πnF t
The harmonic frequencies are nF ∀n and are spaced F = T1 apart.
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 2 / 12
Fourier Series as T → ∞
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
Fourier Series: u(t) =
P∞
n=−∞
Un ei2πnF t
The harmonic frequencies are nF ∀n and are spaced F = T1 apart.
As T gets larger, the harmonic spacing becomes smaller.
e.g. T = 1 s ⇒ F = 1 Hz
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 2 / 12
Fourier Series as T → ∞
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Fourier Series: u(t) =
P∞
n=−∞
Un ei2πnF t
The harmonic frequencies are nF ∀n and are spaced F = T1 apart.
As T gets larger, the harmonic spacing becomes smaller.
e.g. T = 1 s ⇒ F = 1 Hz
T = 1 day ⇒ F = 11.57 µHz
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 2 / 12
Fourier Series as T → ∞
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Fourier Series: u(t) =
P∞
n=−∞
Un ei2πnF t
The harmonic frequencies are nF ∀n and are spaced F = T1 apart.
As T gets larger, the harmonic spacing becomes smaller.
e.g. T = 1 s ⇒ F = 1 Hz
T = 1 day ⇒ F = 11.57 µHz
If T → ∞ then the harmonic spacing becomes zero, the sum becomes an
integral and we get the Fourier Transform:
u(t) =
E1.10 Fourier Series and Transforms (2014-5559)
R +∞
i2πf t
U
(f
)e
df
f =−∞
Fourier Transform: 6 – 2 / 12
Fourier Series as T → ∞
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Fourier Series: u(t) =
P∞
n=−∞
Un ei2πnF t
The harmonic frequencies are nF ∀n and are spaced F = T1 apart.
As T gets larger, the harmonic spacing becomes smaller.
e.g. T = 1 s ⇒ F = 1 Hz
T = 1 day ⇒ F = 11.57 µHz
If T → ∞ then the harmonic spacing becomes zero, the sum becomes an
integral and we get the Fourier Transform:
u(t) =
R +∞
i2πf t
U
(f
)e
df
f =−∞
Here, U (f ), is the spectral density of u(t).
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 2 / 12
Fourier Series as T → ∞
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Fourier Series: u(t) =
P∞
n=−∞
Un ei2πnF t
The harmonic frequencies are nF ∀n and are spaced F = T1 apart.
As T gets larger, the harmonic spacing becomes smaller.
e.g. T = 1 s ⇒ F = 1 Hz
T = 1 day ⇒ F = 11.57 µHz
If T → ∞ then the harmonic spacing becomes zero, the sum becomes an
integral and we get the Fourier Transform:
u(t) =
R +∞
i2πf t
U
(f
)e
df
f =−∞
Here, U (f ), is the spectral density of u(t).
• U (f ) is a continuous function of f .
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 2 / 12
Fourier Series as T → ∞
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Fourier Series: u(t) =
P∞
n=−∞
Un ei2πnF t
The harmonic frequencies are nF ∀n and are spaced F = T1 apart.
As T gets larger, the harmonic spacing becomes smaller.
e.g. T = 1 s ⇒ F = 1 Hz
T = 1 day ⇒ F = 11.57 µHz
If T → ∞ then the harmonic spacing becomes zero, the sum becomes an
integral and we get the Fourier Transform:
u(t) =
R +∞
i2πf t
U
(f
)e
df
f =−∞
Here, U (f ), is the spectral density of u(t).
• U (f ) is a continuous function of f .
• U (f ) is complex-valued.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 2 / 12
Fourier Series as T → ∞
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Fourier Series: u(t) =
P∞
n=−∞
Un ei2πnF t
The harmonic frequencies are nF ∀n and are spaced F = T1 apart.
As T gets larger, the harmonic spacing becomes smaller.
e.g. T = 1 s ⇒ F = 1 Hz
T = 1 day ⇒ F = 11.57 µHz
If T → ∞ then the harmonic spacing becomes zero, the sum becomes an
integral and we get the Fourier Transform:
u(t) =
R +∞
i2πf t
U
(f
)e
df
f =−∞
Here, U (f ), is the spectral density of u(t).
• U (f ) is a continuous function of f .
• U (f ) is complex-valued.
• u(t) real ⇒ U (f ) is conjugate symmetric ⇔ U (−f ) = U (f )∗ .
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 2 / 12
Fourier Series as T → ∞
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Fourier Series: u(t) =
P∞
n=−∞
Un ei2πnF t
The harmonic frequencies are nF ∀n and are spaced F = T1 apart.
As T gets larger, the harmonic spacing becomes smaller.
e.g. T = 1 s ⇒ F = 1 Hz
T = 1 day ⇒ F = 11.57 µHz
If T → ∞ then the harmonic spacing becomes zero, the sum becomes an
integral and we get the Fourier Transform:
u(t) =
R +∞
i2πf t
U
(f
)e
df
f =−∞
Here, U (f ), is the spectral density of u(t).
•
•
•
•
U (f ) is a continuous function of f .
U (f ) is complex-valued.
u(t) real ⇒ U (f ) is conjugate symmetric ⇔ U (−f ) = U (f )∗ .
Units: if u(t) is in volts, then U (f )df must also be in volts
⇒ U (f ) is in volts/Hz (hence “spectral density”).
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 2 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
Fourier Series:
T →∞
• Fourier Transform
• Fourier Transform
u(t) =
P∞
n=−∞
Un ei2πnF t
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Fourier Series:
u(t) =
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
= ∆f
R 0.5T
t=−0.5T
u(t)e−i2πnF t dt
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
= ∆f
R 0.5T
t=−0.5T
u(t)e−i2πnF t dt
Spectral Density: If u(t) has finite energy, Un → 0 as ∆f → 0.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
= ∆f
R 0.5T
t=−0.5T
u(t)e−i2πnF t dt
Spectral Density: If u(t) has finite energy, Un → 0 as ∆f → 0. So we
Un
define a spectral density, U (fn ) = ∆f
, on the set of frequencies {fn }:
U (fn ) =
E1.10 Fourier Series and Transforms (2014-5559)
Un
∆f
=
R 0.5T
t=−0.5T
u(t)e−i2πfn t dt
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
= ∆f
R 0.5T
t=−0.5T
u(t)e−i2πnF t dt
Spectral Density: If u(t) has finite energy, Un → 0 as ∆f → 0. So we
Un
define a spectral density, U (fn ) = ∆f
, on the set of frequencies {fn }:
U (fn ) =
Un
∆f
=
we can write
u(t) =
E1.10 Fourier Series and Transforms (2014-5559)
P∞
n=−∞
R 0.5T
t=−0.5T
u(t)e−i2πfn t dt
[Substitute Un = U (fn )∆f ]
U (fn )ei2πfn t ∆f
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
= ∆f
R 0.5T
t=−0.5T
u(t)e−i2πnF t dt
Spectral Density: If u(t) has finite energy, Un → 0 as ∆f → 0. So we
Un
define a spectral density, U (fn ) = ∆f
, on the set of frequencies {fn }:
U (fn ) =
Un
∆f
=
we can write
u(t) =
P∞
n=−∞
R 0.5T
t=−0.5T
u(t)e−i2πfn t dt
[Substitute Un = U (fn )∆f ]
U (fn )ei2πfn t ∆f
Fourier Transform: Now if we take the limit as ∆f → 0, we get
u(t) =
E1.10 Fourier Series and Transforms (2014-5559)
R∞
i2πf t
U
(f
)e
df
−∞
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
= ∆f
R 0.5T
t=−0.5T
u(t)e−i2πnF t dt
Spectral Density: If u(t) has finite energy, Un → 0 as ∆f → 0. So we
Un
define a spectral density, U (fn ) = ∆f
, on the set of frequencies {fn }:
U (fn ) =
Un
∆f
=
we can write
u(t) =
P∞
n=−∞
R 0.5T
t=−0.5T
u(t)e−i2πfn t dt
[Substitute Un = U (fn )∆f ]
U (fn )ei2πfn t ∆f
Fourier Transform: Now if we take the limit as ∆f → 0, we get
R∞
i2πf t
U
(f
)e
df
−∞
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
u(t) =
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
= ∆f
R 0.5T
t=−0.5T
u(t)e−i2πnF t dt
Spectral Density: If u(t) has finite energy, Un → 0 as ∆f → 0. So we
Un
define a spectral density, U (fn ) = ∆f
, on the set of frequencies {fn }:
U (fn ) =
Un
∆f
=
we can write
u(t) =
P∞
n=−∞
R 0.5T
t=−0.5T
u(t)e−i2πfn t dt
[Substitute Un = U (fn )∆f ]
U (fn )ei2πfn t ∆f
Fourier Transform: Now if we take the limit as ∆f → 0, we get
R∞
i2πf t
U
(f
)e
df
−∞
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
u(t) =
E1.10 Fourier Series and Transforms (2014-5559)
[Fourier Synthesis]
[Fourier Analysis]
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
= ∆f
R 0.5T
t=−0.5T
u(t)e−i2πnF t dt
Spectral Density: If u(t) has finite energy, Un → 0 as ∆f → 0. So we
Un
define a spectral density, U (fn ) = ∆f
, on the set of frequencies {fn }:
U (fn ) =
Un
∆f
=
we can write
u(t) =
P∞
n=−∞
R 0.5T
t=−0.5T
u(t)e−i2πfn t dt
[Substitute Un = U (fn )∆f ]
U (fn )ei2πfn t ∆f
Fourier Transform: Now if we take the limit as ∆f → 0, we get
R∞
i2πf t
U
(f
)e
df
−∞
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
u(t) =
[Fourier Synthesis]
[Fourier Analysis]
Un
For non-periodic signals Un → 0 as ∆f → 0 and U (fn ) = ∆f
remains
finite.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
= ∆f
R 0.5T
t=−0.5T
u(t)e−i2πnF t dt
Spectral Density: If u(t) has finite energy, Un → 0 as ∆f → 0. So we
Un
define a spectral density, U (fn ) = ∆f
, on the set of frequencies {fn }:
U (fn ) =
Un
∆f
=
we can write
u(t) =
P∞
n=−∞
R 0.5T
t=−0.5T
u(t)e−i2πfn t dt
[Substitute Un = U (fn )∆f ]
U (fn )ei2πfn t ∆f
Fourier Transform: Now if we take the limit as ∆f → 0, we get
R∞
i2πf t
U
(f
)e
df
−∞
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
u(t) =
[Fourier Synthesis]
[Fourier Analysis]
Un
For non-periodic signals Un → 0 as ∆f → 0 and U (fn ) = ∆f
remains
finite. However, if u(t) contains an exactly periodic component, then the
corresponding U (fn ) will become infinite as ∆f → 0.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 3 / 12
Fourier Transform
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t) =
Fourier Series:
P∞
n=−∞
Un ei2πnF t
The summation is over a set of equally spaced frequencies
fn = nF where the spacing between them is ∆f = F = T1 .
−i2πnF t
Un = u(t)e
= ∆f
R 0.5T
t=−0.5T
u(t)e−i2πnF t dt
Spectral Density: If u(t) has finite energy, Un → 0 as ∆f → 0. So we
Un
define a spectral density, U (fn ) = ∆f
, on the set of frequencies {fn }:
U (fn ) =
Un
∆f
=
we can write
u(t) =
P∞
n=−∞
R 0.5T
t=−0.5T
u(t)e−i2πfn t dt
[Substitute Un = U (fn )∆f ]
U (fn )ei2πfn t ∆f
Fourier Transform: Now if we take the limit as ∆f → 0, we get
R∞
i2πf t
U
(f
)e
df
−∞
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
u(t) =
[Fourier Synthesis]
[Fourier Analysis]
Un
For non-periodic signals Un → 0 as ∆f → 0 and U (fn ) = ∆f
remains
finite. However, if u(t) contains an exactly periodic component, then the
corresponding U (fn ) will become infinite as ∆f → 0. We will deal with it.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 3 / 12
Fourier Transform Examples
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Example(1:
u(t) =
e−at t ≥ 0
0
t<0
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Example(1:
u(t) =
e−at t ≥ 0
0
t<0
1
u(t)
6: Fourier Transform
• Fourier Series as
a=2
0.5
0
-5
0
Time (s)
5
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example(1:
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
E1.10 Fourier Series and Transforms (2014-5559)
1
u(t)
6: Fourier Transform
• Fourier Series as
a=2
0.5
0
-5
0
Time (s)
5
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example(1:
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
E1.10 Fourier Series and Transforms (2014-5559)
1
u(t)
6: Fourier Transform
• Fourier Series as
a=2
0.5
0
-5
0
Time (s)
5
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example(1:
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
E1.10 Fourier Series and Transforms (2014-5559)
1
u(t)
6: Fourier Transform
• Fourier Series as
a=2
0.5
0
-5
0
Time (s)
5
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example(1:
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
E1.10 Fourier Series and Transforms (2014-5559)
1
u(t)
6: Fourier Transform
• Fourier Series as
a=2
0.5
0
-5
0
Time (s)
5
1
a+i2πf
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
E1.10 Fourier Series and Transforms (2014-5559)
1
u(t)
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
a=2
0.5
0
|U(f)|
6: Fourier Transform
• Fourier Series as
0.5
0.4
0.3
0.2
0.1
-5
-5
0
Time (s)
0
Frequency (Hz)
5
5
1
a+i2πf
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
u(t)
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
1
a=2
0.5
0
|U(f)|
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
<U(f) (rad/pi)
6: Fourier Transform
• Fourier Series as
0.5
0.4
0.3
0.2
0.1
-5
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
a+i2πf
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t)
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
1
a=2
0.5
0
|U(f)|
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
<U(f) (rad/pi)
6: Fourier Transform
• Fourier Series as
0.5
0.4
0.3
0.2
0.1
-5
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
a+i2πf
Example 2:
v(t) = e−a|t|
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example 2:
v(t) = e−a|t|
E1.10 Fourier Series and Transforms (2014-5559)
u(t)
a=2
0.5
0
|U(f)|
Examples
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
1
<U(f) (rad/pi)
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
a+i2πf
1
v(t)
6: Fourier Transform
• Fourier Series as
a=2
0.5
0
-5
0
Time (s)
5
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example 2:
v(t) = e−a|t|
R∞
V (f ) = −∞ v(t)e−i2πf t dt
E1.10 Fourier Series and Transforms (2014-5559)
u(t)
a=2
0.5
0
|U(f)|
Examples
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
1
<U(f) (rad/pi)
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
a+i2πf
1
v(t)
6: Fourier Transform
• Fourier Series as
a=2
0.5
0
-5
0
Time (s)
5
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example 2:
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
u(t)
1
a+i2πf
1
a=2
0.5
0
v(t) = e−a|t|
R∞
V (f ) = −∞ v(t)e−i2πf t dt
R ∞ −at −i2πf t
R0
at −i2πf t
dt
dt + 0 e e
= −∞ e e
E1.10 Fourier Series and Transforms (2014-5559)
a=2
0.5
0
|U(f)|
Examples
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
1
<U(f) (rad/pi)
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
v(t)
6: Fourier Transform
• Fourier Series as
-5
0
Time (s)
5
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example 2:
u(t)
a=2
0.5
0
|U(f)|
Examples
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
1
<U(f) (rad/pi)
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
a=2
0.5
0
-5
0
Time (s)
v(t) = e−a|t|
R∞
V (f ) = −∞ v(t)e−i2πf t dt
R ∞ −at −i2πf t
R0
at −i2πf t
dt
dt + 0 e e
= −∞ e e
(a−i2πf )t 0
(−a−i2πf )t ∞
−1
1
+ a+i2πf e
= a−i2πf e
−∞
0
E1.10 Fourier Series and Transforms (2014-5559)
5
1
a+i2πf
v(t)
6: Fourier Transform
• Fourier Series as
5
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example 2:
u(t)
a=2
0.5
0
|U(f)|
Examples
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
1
<U(f) (rad/pi)
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
a=2
0.5
0
-5
0
Time (s)
v(t) = e−a|t|
R∞
V (f ) = −∞ v(t)e−i2πf t dt
R ∞ −at −i2πf t
R0
at −i2πf t
dt
dt + 0 e e
= −∞ e e
(a−i2πf )t 0
(−a−i2πf )t ∞
−1
1
+ a+i2πf e
= a−i2πf e
−∞
0
1
1
= a−i2πf + a+i2πf
E1.10 Fourier Series and Transforms (2014-5559)
5
1
a+i2πf
v(t)
6: Fourier Transform
• Fourier Series as
5
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example 2:
u(t)
a=2
0.5
0
|U(f)|
Examples
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
1
<U(f) (rad/pi)
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
a=2
0.5
0
-5
0
Time (s)
v(t) = e−a|t|
R∞
V (f ) = −∞ v(t)e−i2πf t dt
R ∞ −at −i2πf t
R0
at −i2πf t
dt
dt + 0 e e
= −∞ e e
(a−i2πf )t 0
(−a−i2πf )t ∞
−1
1
+ a+i2πf e
= a−i2πf e
−∞
0
1
1
2a
= a−i2πf + a+i2πf = a2 +4π2 f 2
E1.10 Fourier Series and Transforms (2014-5559)
5
1
a+i2πf
v(t)
6: Fourier Transform
• Fourier Series as
5
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example 2:
u(t)
a=2
0.5
0
|U(f)|
Examples
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
1
<U(f) (rad/pi)
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
a+i2πf
1
v(t)
6: Fourier Transform
• Fourier Series as
a=2
0.5
0
-5
0
5
|V(f)|
Time (s)
v(t) = e−a|t|
1
R∞
0.5
V (f ) = −∞ v(t)e−i2πf t dt
0
-5
0
Frequency (Hz)
R ∞ −at −i2πf t
R0
at −i2πf t
dt
dt + 0 e e
= −∞ e e
(a−i2πf )t 0
(−a−i2πf )t ∞
−1
1
+ a+i2πf e
= a−i2πf e
−∞
0
1
1
2a
= a−i2πf + a+i2πf = a2 +4π2 f 2
E1.10 Fourier Series and Transforms (2014-5559)
5
Fourier Transform: 6 – 4 / 12
Fourier Transform Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Example 2:
u(t)
a=2
0.5
0
|U(f)|
Examples
e−at t ≥ 0
u(t) =
0
t<0
R∞
U (f ) = −∞ u(t)e−i2πf t dt
R ∞ −at −i2πf t
dt
= 0 e e
R ∞ (−a−i2πf )t
dt
= 0 e
(−a−i2πf )t ∞
−1
= a+i2πf e
=
0
1
<U(f) (rad/pi)
T →∞
• Fourier Transform
• Fourier Transform
Example(1:
-5
0.5
0.4
0.3
0.2
0.1
-5
0
Time (s)
5
0
Frequency (Hz)
5
0
Frequence (Hz)
5
0.5
0
-0.5
-5
1
a+i2πf
1
v(t)
6: Fourier Transform
• Fourier Series as
a=2
0.5
0
-5
0
5
|V(f)|
Time (s)
v(t) = e−a|t|
1
R∞
0.5
V (f ) = −∞ v(t)e−i2πf t dt
0
-5
0
5
Frequency (Hz)
R ∞ −at −i2πf t
R0
at −i2πf t
dt
dt + 0 e e
= −∞ e e
(a−i2πf )t 0
(−a−i2πf )t ∞
−1
1
+ a+i2πf e
= a−i2πf e
−∞
0
1
1
2a
= a−i2πf + a+i2πf = a2 +4π2 f 2
[v(t) real+symmetric
⇒ V (f ) real+symmetric]
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 4 / 12
Dirac Delta Function
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
We define a unit area pulse of width w as
dw (x) =
(
1
w
0
−0.5w ≤ x ≤ 0.5w
otherwise
4
2
δ 3(x)
0
-3
-2
-1
0
x
1
2
3
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 5 / 12
Dirac Delta Function
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
We define a unit area pulse of width w as
dw (x) =
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
(
1
w
0
−0.5w ≤ x ≤ 0.5w
otherwise
4
2
δ 3(x)
0
-3
-2
-1
0
x
1
2
3
This pulse has the property that its integral equals
1 for all values of w:
R∞
x=−∞
E1.10 Fourier Series and Transforms (2014-5559)
dw (x)dx = 1
Fourier Transform: 6 – 5 / 12
Dirac Delta Function
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
We define a unit area pulse of width w as
dw (x) =
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
(
1
w
0
−0.5w ≤ x ≤ 0.5w
otherwise
4
δ 0.5(x)
2
δ 3(x)
0
-3
-2
-1
0
x
1
2
3
This pulse has the property that its integral equals
1 for all values of w:
R∞
x=−∞
dw (x)dx = 1
If we make w smaller, the pulse height increases to preserve unit area.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 5 / 12
Dirac Delta Function
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
We define a unit area pulse of width w as
dw (x) =
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
(
1
w
0
−0.5w ≤ x ≤ 0.5w
otherwise
δ 0.2(x)
4
δ 0.5(x)
2
δ 3(x)
0
-3
-2
-1
0
x
1
2
3
This pulse has the property that its integral equals
1 for all values of w:
R∞
x=−∞
dw (x)dx = 1
If we make w smaller, the pulse height increases to preserve unit area.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 5 / 12
Dirac Delta Function
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
We define a unit area pulse of width w as
dw (x) =
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
(
1
w
0
−0.5w ≤ x ≤ 0.5w
otherwise
δ 0.2(x)
4
δ 0.5(x)
2
δ 3(x)
0
-3
-2
-1
0
x
1
2
3
This pulse has the property that its integral equals
1 for all values of w:
R∞
x=−∞
dw (x)dx = 1
If we make w smaller, the pulse height increases to preserve unit area.
We define the Dirac delta function as δ(x) = limw→0 dw (x)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 5 / 12
Dirac Delta Function
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
We define a unit area pulse of width w as
dw (x) =
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
(
1
w
0
−0.5w ≤ x ≤ 0.5w
otherwise
δ 0.2(x)
4
δ 0.5(x)
2
δ 3(x)
0
-3
-2
-1
0
x
1
2
3
This pulse has the property that its integral equals
1 for all values of w:
R∞
x=−∞
dw (x)dx = 1
If we make w smaller, the pulse height increases to preserve unit area.
We define the Dirac delta function as δ(x) = limw→0 dw (x)
• δ(x) equals zero everywhere except at x = 0 where it is infinite.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 5 / 12
Dirac Delta Function
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
We define a unit area pulse of width w as
dw (x) =
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
(
1
w
0
−0.5w ≤ x ≤ 0.5w
otherwise
δ 0.2(x)
4
δ 0.5(x)
2
δ 3(x)
0
-3
-2
-1
0
x
1
2
3
This pulse has the property that its integral equals
1 for all values of w:
R∞
x=−∞
dw (x)dx = 1
If we make w smaller, the pulse height increases to preserve unit area.
We define the Dirac delta function as δ(x) = limw→0 dw (x)
• δ(x) equals zero everywhere except at x = 0 where it is infinite.
R∞
• However its area still equals 1 ⇒ −∞ δ(x)dx = 1
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 5 / 12
Dirac Delta Function
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
We define a unit area pulse of width w as
dw (x) =
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
(
1
w
0
−0.5w ≤ x ≤ 0.5w
otherwise
This pulse has the property that its integral equals
1 for all values of w:
R∞
x=−∞
dw (x)dx = 1
δ 0.2(x)
4
δ 0.5(x)
2
δ 3(x)
0
-3
-2
-1
0
x
1
2
3
1
2
3
1
δ (x)
0.5
0
-3
-2
-1
0
x
If we make w smaller, the pulse height increases to preserve unit area.
We define the Dirac delta function as δ(x) = limw→0 dw (x)
• δ(x) equals zero everywhere except at x = 0 where it is infinite.
R∞
• However its area still equals 1 ⇒ −∞ δ(x)dx = 1
• We plot the height of δ(x) as its area rather than its true height of ∞.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 5 / 12
Dirac Delta Function
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
We define a unit area pulse of width w as
dw (x) =
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
(
1
w
0
−0.5w ≤ x ≤ 0.5w
otherwise
This pulse has the property that its integral equals
1 for all values of w:
R∞
x=−∞
dw (x)dx = 1
δ 0.2(x)
4
δ 0.5(x)
2
δ 3(x)
0
-3
-2
-1
0
x
1
2
3
1
2
3
1
δ (x)
0.5
0
-3
-2
-1
0
x
If we make w smaller, the pulse height increases to preserve unit area.
We define the Dirac delta function as δ(x) = limw→0 dw (x)
• δ(x) equals zero everywhere except at x = 0 where it is infinite.
R∞
• However its area still equals 1 ⇒ −∞ δ(x)dx = 1
• We plot the height of δ(x) as its area rather than its true height of ∞.
δ(x) is not quite a proper function: it is called a generalized function.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 5 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ (x)
1
0
Examples
• Dirac Delta Function
• Dirac Delta Function:
-1
-3
-2
-1
0
x
1
2
3
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
• Dirac Delta Function
• Dirac Delta Function:
δ (x)
1
δ (x-2)
0
-1
-3
-2
-1
0
x
1
2
3
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
δ (x-2)
0
-1
-3
Amplitude Scaling: bδ(x)
δ (x)
1
-2
-1
0
x
1
2
3
δ(x) has an area of 1
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
Amplitude Scaling: bδ(x)
E1.10 Fourier Series and Transforms (2014-5559)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
(bδ(x)) dx
−∞
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx
−∞
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
c > 0:
R∞
x=−∞
E1.10 Fourier Series and Transforms (2014-5559)
δ(cx)dx
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
c > 0:
R∞
δ(cx)dx =
x=−∞
E1.10 Fourier Series and Transforms (2014-5559)
R∞
dy
δ(y)
c
y=−∞
[sub y = cx]
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
R∞
c > 0: x=−∞ δ(cx)dx = y=−∞ δ(y) dy
c
R
∞
= 1c y=−∞ δ(y)dy
R∞
E1.10 Fourier Series and Transforms (2014-5559)
[sub y = cx]
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
R∞
c > 0: x=−∞ δ(cx)dx = y=−∞ δ(y) dy
c
R
∞
= 1c y=−∞ δ(y)dy =
R∞
E1.10 Fourier Series and Transforms (2014-5559)
[sub y = cx]
1
c
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
R∞
c > 0: x=−∞ δ(cx)dx = y=−∞ δ(y) dy
c
R
∞
= 1c y=−∞ δ(y)dy =
R∞
c < 0: x=−∞ δ(cx)dx
R∞
E1.10 Fourier Series and Transforms (2014-5559)
[sub y = cx]
1
c
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
R∞
c > 0: x=−∞ δ(cx)dx = y=−∞ δ(y) dy
c
R
∞
= 1c y=−∞ δ(y)dy =
R −∞
R∞
c < 0: x=−∞ δ(cx)dx = y=+∞ δ(y) dy
c
R∞
E1.10 Fourier Series and Transforms (2014-5559)
[sub y = cx]
1
c
[sub y = cx]
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
R∞
c > 0: x=−∞ δ(cx)dx = y=−∞ δ(y) dy
c
R
∞
= 1c y=−∞ δ(y)dy =
R −∞
R∞
c < 0: x=−∞ δ(cx)dx = y=+∞ δ(y) dy
c
R
+∞
δ(y)dy
= −1
c
y=−∞
R∞
E1.10 Fourier Series and Transforms (2014-5559)
[sub y = cx]
1
c
[sub y = cx]
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
-0.5δ(x+2)
Amplitude Scaling: bδ(x)
R∞
−∞
δ (x-2)
0
-1
-3
δ(x) has an area of 1 ⇔
δ (x)
1
-2
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
R∞
c > 0: x=−∞ δ(cx)dx = y=−∞ δ(y) dy
c
R
∞
= 1c y=−∞ δ(y)dy = 1c
R −∞
R∞
c < 0: x=−∞ δ(cx)dx = y=+∞ δ(y) dy
c
R
+∞
−1
δ(y)dy
=
= −1
c
c
y=−∞
R∞
E1.10 Fourier Series and Transforms (2014-5559)
[sub y = cx]
[sub y = cx]
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ (x)
1
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
0
-0.5δ(x+2)
-1
-3
-2
Amplitude Scaling: bδ(x)
δ(x) has an area of 1 ⇔
R∞
−∞
δ (x-2)
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
R∞
c > 0: x=−∞ δ(cx)dx = y=−∞ δ(y) dy
c
R
∞
1
= 1c y=−∞ δ(y)dy = 1c = |c|
R −∞
R∞
c < 0: x=−∞ δ(cx)dx = y=+∞ δ(y) dy
c
R
+∞
−1
δ(y)dy
=
= −1
c
c =
y=−∞
R∞
E1.10 Fourier Series and Transforms (2014-5559)
[sub y = cx]
[sub y = cx]
1
|c|
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ (x)
1
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
0
-0.5δ(x+2)
-1
-3
-2
Amplitude Scaling: bδ(x)
δ(x) has an area of 1 ⇔
R∞
−∞
δ (x-2)
-1
0
x
1
2
3
δ(x)dx = 1
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
R∞
c > 0: x=−∞ δ(cx)dx = y=−∞ δ(y) dy
c
R
∞
1
= 1c y=−∞ δ(y)dy = 1c = |c|
R −∞
R∞
c < 0: x=−∞ δ(cx)dx = y=+∞ δ(y) dy
c
R
+∞
−1
δ(y)dy
=
= −1
c
c =
y=−∞
R∞
1
In general, δ(cx) = |c|
δ(x) for c 6= 0
E1.10 Fourier Series and Transforms (2014-5559)
[sub y = cx]
[sub y = cx]
1
|c|
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ (x)
1
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
0
-0.5δ(x+2)
-1
-3
-2
-1
Amplitude Scaling: bδ(x)
δ(x) has an area of 1 ⇔
R∞
δ(x)dx = 1
−∞
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
δ (x-2)
0
x
1
2
3
1
δ (4x) = 0.25δ (x)
0
-1
-3
-2
-1
0
x
1
2
3
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
R∞
c > 0: x=−∞ δ(cx)dx = y=−∞ δ(y) dy
c
R
∞
1
= 1c y=−∞ δ(y)dy = 1c = |c|
R −∞
R∞
c < 0: x=−∞ δ(cx)dx = y=+∞ δ(y) dy
c
R
+∞
−1
δ(y)dy
=
= −1
c
c =
y=−∞
R∞
1
In general, δ(cx) = |c|
δ(x) for c 6= 0
E1.10 Fourier Series and Transforms (2014-5559)
[sub y = cx]
[sub y = cx]
1
|c|
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Scaling and Translation
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
Translation: δ(x − a)
δ (x)
1
δ(x) is a pulse at x = 0
δ(x − a) is a pulse at x = a
0
-0.5δ(x+2)
-1
-3
-2
Amplitude Scaling: bδ(x)
δ(x) has an area of 1 ⇔
R∞
δ(x)dx = 1
−∞
bδ(x)
has an area of b since
R∞
R∞
(bδ(x)) dx = b −∞ δ(x)dx = b
−∞
δ (x-2)
-1
0
x
1
2
3
1
δ (4x) = 0.25δ (x)
0
-3δ(-4x-8) = -0.75δ(x+2)
-2
-1
0
1
2
x
-1
-3
3
b can be a complex number (on a graph, we then plot only its magnitude)
Time Scaling: δ(cx)
R∞
c > 0: x=−∞ δ(cx)dx = y=−∞ δ(y) dy
c
R
∞
1
= 1c y=−∞ δ(y)dy = 1c = |c|
R −∞
R∞
c < 0: x=−∞ δ(cx)dx = y=+∞ δ(y) dy
c
R
+∞
−1
δ(y)dy
=
= −1
c
c =
y=−∞
R∞
1
In general, δ(cx) = |c|
δ(x) for c 6= 0
E1.10 Fourier Series and Transforms (2014-5559)
[sub y = cx]
[sub y = cx]
1
|c|
Fourier Transform: 6 – 6 / 12
Dirac Delta Function: Products and Integrals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
If we multiply δ(x − a) by a function of x:
y = x2 × δ(x − 2)
6
y=x2
4
2
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
0
-1
0
1
2
1
2
x
6
y=δ(x-2)
4
2
0
-1
0
x
Fourier Transform: 6 – 7 / 12
Dirac Delta Function: Products and Integrals
6: Fourier Transform
• Fourier Series as
If we multiply δ(x − a) by a function of x:
T →∞
• Fourier Transform
• Fourier Transform
y = x2 × δ(x − 2)
The product is 0 everywhere except at x = 2.
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
6
y=x2
4
2
0
-1
0
1
2
1
2
x
6
y=δ(x-2)
4
2
0
-1
0
x
Fourier Transform: 6 – 7 / 12
Dirac Delta Function: Products and Integrals
6: Fourier Transform
• Fourier Series as
If we multiply δ(x − a) by a function of x:
T →∞
• Fourier Transform
• Fourier Transform
y = x2 × δ(x − 2)
The product is 0 everywhere except at x = 2.
So δ(x − 2) is multiplied by the value taken by
x2 at x = 2:
2
2
x × δ(x − 2) = x x=2 × δ(x − 2)
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
6
y=x2
4
2
0
-1
0
1
2
1
2
x
6
y=δ(x-2)
4
2
0
-1
0
x
Fourier Transform: 6 – 7 / 12
Dirac Delta Function: Products and Integrals
6: Fourier Transform
• Fourier Series as
If we multiply δ(x − a) by a function of x:
T →∞
• Fourier Transform
• Fourier Transform
y = x2 × δ(x − 2)
The product is 0 everywhere except at x = 2.
So δ(x − 2) is multiplied by the value taken by
x2 at x = 2:
2
2
x × δ(x − 2) = x x=2 × δ(x − 2)
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
= 4 × δ(x − 2)
6
y=x2
4
2
0
-1
0
1
2
1
2
x
6
y=δ(x-2)
4
2
0
-1
0
x
6
y=22×δ(x-2) = 4δ(x-2)
4
2
0
-1
0
1
2
x
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 7 / 12
Dirac Delta Function: Products and Integrals
6: Fourier Transform
• Fourier Series as
If we multiply δ(x − a) by a function of x:
T →∞
• Fourier Transform
• Fourier Transform
y = x2 × δ(x − 2)
The product is 0 everywhere except at x = 2.
So δ(x − 2) is multiplied by the value taken by
x2 at x = 2:
2
2
x × δ(x − 2) = x x=2 × δ(x − 2)
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
= 4 × δ(x − 2)
6
y=x2
4
2
0
-1
0
1
2
1
2
x
6
y=δ(x-2)
4
2
0
-1
0
x
6
y=22×δ(x-2) = 4δ(x-2)
4
2
0
-1
0
1
2
x
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 7 / 12
Dirac Delta Function: Products and Integrals
6: Fourier Transform
• Fourier Series as
If we multiply δ(x − a) by a function of x:
T →∞
• Fourier Transform
• Fourier Transform
y = x2 × δ(x − 2)
The product is 0 everywhere except at x = 2.
So δ(x − 2) is multiplied by the value taken by
x2 at x = 2:
2
2
x × δ(x − 2) = x x=2 × δ(x − 2)
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
= 4 × δ(x − 2)
6
y=x2
4
2
0
-1
0
1
2
1
2
x
6
y=δ(x-2)
4
2
0
-1
0
x
In general for any function, f (x), that is
continuous at x = a,
f (x)δ(x − a) = f (a)δ(x − a)
6
y=22×δ(x-2) = 4δ(x-2)
4
2
0
-1
0
1
2
x
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 7 / 12
Dirac Delta Function: Products and Integrals
6: Fourier Transform
• Fourier Series as
If we multiply δ(x − a) by a function of x:
T →∞
• Fourier Transform
• Fourier Transform
y = x2 × δ(x − 2)
The product is 0 everywhere except at x = 2.
So δ(x − 2) is multiplied by the value taken by
x2 at x = 2:
2
2
x × δ(x − 2) = x x=2 × δ(x − 2)
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
6
y=x2
4
2
0
-1
0
1
2
1
2
x
6
y=δ(x-2)
4
2
= 4 × δ(x − 2)
0
-1
0
x
In general for any function, f (x), that is
continuous at x = a,
6
4
2
f (x)δ(x − a) = f (a)δ(x − a)
0
-1
−∞
0
1
2
x
Integrals:
R∞
y=22×δ(x-2) = 4δ(x-2)
f (x)δ(x − a)dx =
E1.10 Fourier Series and Transforms (2014-5559)
R∞
−∞
f (a)δ(x − a)dx
[if f (x) continuous at a]
Fourier Transform: 6 – 7 / 12
Dirac Delta Function: Products and Integrals
6: Fourier Transform
• Fourier Series as
If we multiply δ(x − a) by a function of x:
T →∞
• Fourier Transform
• Fourier Transform
y = x2 × δ(x − 2)
The product is 0 everywhere except at x = 2.
So δ(x − 2) is multiplied by the value taken by
x2 at x = 2:
2
2
x × δ(x − 2) = x x=2 × δ(x − 2)
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
= 4 × δ(x − 2)
6
y=x2
4
2
0
-1
0
1
2
1
2
x
6
y=δ(x-2)
4
2
0
-1
0
x
In general for any function, f (x), that is
continuous at x = a,
f (x)δ(x − a) = f (a)δ(x − a)
−∞
y=22×δ(x-2) = 4δ(x-2)
4
2
0
-1
0
1
2
x
Integrals:
R∞
6
R∞
f (x)δ(x − a)dx = −∞ f (a)δ(x − a)dx
R∞
= f (a) −∞ δ(x − a)dx
[if f (x) continuous at a]
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 7 / 12
Dirac Delta Function: Products and Integrals
6: Fourier Transform
• Fourier Series as
If we multiply δ(x − a) by a function of x:
T →∞
• Fourier Transform
• Fourier Transform
y = x2 × δ(x − 2)
The product is 0 everywhere except at x = 2.
So δ(x − 2) is multiplied by the value taken by
x2 at x = 2:
2
2
x × δ(x − 2) = x x=2 × δ(x − 2)
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
= 4 × δ(x − 2)
6
y=x2
4
2
0
-1
0
1
2
1
2
x
6
y=δ(x-2)
4
2
0
-1
0
x
In general for any function, f (x), that is
continuous at x = a,
f (x)δ(x − a) = f (a)δ(x − a)
−∞
y=22×δ(x-2) = 4δ(x-2)
4
2
0
-1
0
1
2
x
Integrals:
R∞
6
R∞
f (x)δ(x − a)dx = −∞ f (a)δ(x − a)dx
R∞
= f (a) −∞ δ(x − a)dx
= f (a)
[if f (x) continuous at a]
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 7 / 12
Dirac Delta Function: Products and Integrals
6: Fourier Transform
• Fourier Series as
If we multiply δ(x − a) by a function of x:
T →∞
• Fourier Transform
• Fourier Transform
y = x2 × δ(x − 2)
The product is 0 everywhere except at x = 2.
So δ(x − 2) is multiplied by the value taken by
x2 at x = 2:
2
2
x × δ(x − 2) = x x=2 × δ(x − 2)
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
= 4 × δ(x − 2)
6
y=x2
4
2
0
-1
0
1
2
1
2
x
6
y=δ(x-2)
4
2
0
-1
0
x
In general for any function, f (x), that is
continuous at x = a,
f (x)δ(x − a) = f (a)δ(x − a)
y=22×δ(x-2) = 4δ(x-2)
4
2
0
-1
0
1
2
x
Integrals:
R∞
6
R∞
f (x)δ(x − a)dx = −∞ f (a)δ(x − a)dx
R∞
= f (a) −∞ δ(x − a)dx
= f (a)
[if f (x) continuous at a]
2
R∞
2
Example: −∞ 3x − 2x δ(x − 2)dx = 3x − 2x x=2 = 8
−∞
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 7 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
[Fourier Synthesis]
[Fourier Analysis]
1.5
1.5δ(f+2)
1.5δ(f-2)
1
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
u(t) =
R∞
−∞
E1.10 Fourier Series and Transforms (2014-5559)
U (f )ei2πf t df
[Fourier Synthesis]
[Fourier Analysis]
1.5
1.5δ(f+2)
1.5δ(f-2)
1
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
E1.10 Fourier Series and Transforms (2014-5559)
[Fourier Synthesis]
[Fourier Analysis]
1.5
1.5δ(f+2)
1.5δ(f-2)
1
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
i2πf t i2πf t + 1.5 e
= 1.5 e
f =+2
f =−2
E1.10 Fourier Series and Transforms (2014-5559)
[Fourier Synthesis]
[Fourier Analysis]
1.5
1.5δ(f+2)
1.5δ(f-2)
1
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
i2πf t i2πf t + 1.5 e
= 1.5 e
f =+2
f =−2 = 1.5 ei4πt + e−i4πt
E1.10 Fourier Series and Transforms (2014-5559)
[Fourier Synthesis]
[Fourier Analysis]
1.5
1.5δ(f+2)
1.5δ(f-2)
1
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
i2πf t i2πf t + 1.5 e
= 1.5 e
f =+2
f =−2 = 1.5 ei4πt + e−i4πt = 3 cos 4πt
E1.10 Fourier Series and Transforms (2014-5559)
[Fourier Synthesis]
[Fourier Analysis]
1.5
1.5δ(f+2)
1.5δ(f-2)
1
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
i2πf t i2πf t + 1.5 e
= 1.5 e
f =+2
f =−2 = 1.5 ei4πt + e−i4πt = 3 cos 4πt
E1.10 Fourier Series and Transforms (2014-5559)
[Fourier Synthesis]
[Fourier Analysis]
1.5
1.5δ(f+2)
1.5δ(f-2)
1
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
4 3cos(4πt)
2
0
-2
0
1
2
3
Time (s)
4
5
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
i2πf t i2πf t + 1.5 e
= 1.5 e
f =+2
f =−2 = 1.5 ei4πt + e−i4πt = 3 cos 4πt
[Fourier Synthesis]
[Fourier Analysis]
1.5
1.5δ(f+2)
1.5δ(f-2)
1
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
4 3cos(4πt)
2
0
-2
0
1
2
3
Time (s)
4
5
If u(t) is periodic then U (f ) is a sum of Dirac delta functions:
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
1.5
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
1.5δ(f+2)
1.5δ(f-2)
1
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
i2πf t i2πf t + 1.5 e
= 1.5 e
f =+2
f =−2 = 1.5 ei4πt + e−i4πt = 3 cos 4πt
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
4 3cos(4πt)
2
0
-2
0
1
2
3
Time (s)
4
5
If u(t) is periodic then U (f ) is a sum of Dirac delta functions:
u(t) =
P∞
n=−∞
E1.10 Fourier Series and Transforms (2014-5559)
i2πnF t
Un e
⇒ U (f ) =
P∞
n=−∞
Un δ (f − nF )
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
1.5
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
1.5δ(f+2)
1.5δ(f-2)
1
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
i2πf t i2πf t + 1.5 e
= 1.5 e
f =+2
f =−2 = 1.5 ei4πt + e−i4πt = 3 cos 4πt
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
4 3cos(4πt)
2
0
-2
0
1
2
3
Time (s)
4
5
If u(t) is periodic then U (f ) is a sum of Dirac delta functions:
u(t) =
P∞
n=−∞
i2πnF t
Un e
R∞
⇒ U (f ) =
Proof: u(t) = −∞ U (f )ei2πf t df
E1.10 Fourier Series and Transforms (2014-5559)
P∞
n=−∞
Un δ (f − nF )
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
1.5
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
1.5δ(f+2)
1.5δ(f-2)
1
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
i2πf t i2πf t + 1.5 e
= 1.5 e
f =+2
f =−2 = 1.5 ei4πt + e−i4πt = 3 cos 4πt
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
4 3cos(4πt)
2
0
-2
0
1
2
3
Time (s)
4
5
If u(t) is periodic then U (f ) is a sum of Dirac delta functions:
u(t) =
P∞
i2πnF t
n=−∞
Un e
R∞
⇒ U (f ) =
Proof: u(t) = −∞ U (f )ei2πf t df
=
E1.10 Fourier Series and Transforms (2014-5559)
P∞
n=−∞
Un δ (f − nF )
R ∞ P∞
−∞
i2πf t
U
δ
(f
−
nF
)
e
df
n
n=−∞
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
1.5
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
1.5δ(f+2)
1.5δ(f-2)
1
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
i2πf t i2πf t + 1.5 e
= 1.5 e
f =+2
f =−2 = 1.5 ei4πt + e−i4πt = 3 cos 4πt
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
4 3cos(4πt)
2
0
-2
0
1
2
3
Time (s)
4
5
If u(t) is periodic then U (f ) is a sum of Dirac delta functions:
u(t) =
P∞
i2πnF t
n=−∞
Un e
R∞
⇒ U (f ) =
Proof: u(t) = −∞ U (f )ei2πf t df
P∞
n=−∞
Un δ (f − nF )
R ∞ P∞
i2πf t
U
δ
(f
−
nF
)
e
df
n
n=−∞
−∞
R∞
P∞
= n=−∞ Un −∞ δ (f − nF ) ei2πf t df
=
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 8 / 12
Periodic Signals
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
1.5
Example: U (f ) = 1.5δ(f + 2) + 1.5δ(f − 2)
1.5δ(f+2)
1.5δ(f-2)
1
R∞
u(t) = −∞ U (f )ei2πf t df
R∞
= −∞ 1.5δ(f + 2)ei2πf t df
R∞
+ −∞ 1.5δ(f − 2)ei2πf t df
i2πf t i2πf t + 1.5 e
= 1.5 e
f =+2
f =−2 = 1.5 ei4πt + e−i4πt = 3 cos 4πt
0.5
0
-3
-2
-1
0
1
Frequency (Hz)
2
3
4 3cos(4πt)
2
0
-2
0
1
2
3
Time (s)
4
5
If u(t) is periodic then U (f ) is a sum of Dirac delta functions:
u(t) =
P∞
i2πnF t
n=−∞
Un e
R∞
⇒ U (f ) =
Proof: u(t) = −∞ U (f )ei2πf t df
n=−∞
Un δ (f − nF )
R ∞ P∞
i2πf t
U
δ
(f
−
nF
)
e
df
n
n=−∞
−∞
R∞
P∞
= n=−∞ Un −∞ δ (f − nF ) ei2πf t df
P∞
= n=−∞ Un ei2πnF t
=
E1.10 Fourier Series and Transforms (2014-5559)
P∞
Fourier Transform: 6 – 8 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t)
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t), then
V (f ) =
E1.10 Fourier Series and Transforms (2014-5559)
R∞
−i2πf t
v(t)e
dτ
t=−∞
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t), then
R∞
V (f ) = t=−∞ v(t)e−i2πf t dτ
R∞
V (g) = t=−∞ U (t)e−i2πgt dt
E1.10 Fourier Series and Transforms (2014-5559)
[substitute f = g , v(t) = U (t)]
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t), then
R∞
V (f ) = t=−∞ v(t)e−i2πf t dτ
R∞
V (g) = t=−∞ U (t)e−i2πgt dt [substitute f = g , v(t) = U (t)]
R∞
= f =−∞ U (f )e−i2πgf df
[substitute t = f ]
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t), then
R∞
V (f ) = t=−∞ v(t)e−i2πf t dτ
R∞
V (g) = t=−∞ U (t)e−i2πgt dt [substitute f = g , v(t) = U (t)]
R∞
= f =−∞ U (f )e−i2πgf df
[substitute t = f ]
= u(−g)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t), then
R∞
V (f ) = t=−∞ v(t)e−i2πf t dτ
R∞
V (g) = t=−∞ U (t)e−i2πgt dt [substitute f = g , v(t) = U (t)]
R∞
= f =−∞ U (f )e−i2πgf df
[substitute t = f ]
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
So:
= u(−g)
v(t) = U (t) ⇒ V (f ) = u(−f )
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t), then
R∞
V (f ) = t=−∞ v(t)e−i2πf t dτ
R∞
V (g) = t=−∞ U (t)e−i2πgt dt [substitute f = g , v(t) = U (t)]
R∞
= f =−∞ U (f )e−i2πgf df
[substitute t = f ]
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
So:
= u(−g)
v(t) = U (t) ⇒ V (f ) = u(−f )
Example:
u(t) = e−|t|
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t), then
R∞
V (f ) = t=−∞ v(t)e−i2πf t dτ
R∞
V (g) = t=−∞ U (t)e−i2πgt dt [substitute f = g , v(t) = U (t)]
R∞
= f =−∞ U (f )e−i2πgf df
[substitute t = f ]
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
So:
= u(−g)
v(t) = U (t) ⇒ V (f ) = u(−f )
Example:
u(t) = e−|t|
E1.10 Fourier Series and Transforms (2014-5559)
⇒
U (f ) =
2
1+4π 2 f 2
[from earlier]
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t), then
R∞
V (f ) = t=−∞ v(t)e−i2πf t dτ
R∞
V (g) = t=−∞ U (t)e−i2πgt dt [substitute f = g , v(t) = U (t)]
R∞
= f =−∞ U (f )e−i2πgf df
[substitute t = f ]
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
So:
= u(−g)
v(t) = U (t) ⇒ V (f ) = u(−f )
Example:
u(t) = e−|t|
v(t) =
⇒
U (f ) =
2
1+4π 2 f 2
[from earlier]
2
1+4π 2 t2
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t), then
R∞
V (f ) = t=−∞ v(t)e−i2πf t dτ
R∞
V (g) = t=−∞ U (t)e−i2πgt dt [substitute f = g , v(t) = U (t)]
R∞
= f =−∞ U (f )e−i2πgf df
[substitute t = f ]
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
So:
= u(−g)
v(t) = U (t) ⇒ V (f ) = u(−f )
Example:
u(t) = e−|t|
v(t) =
2
1+4π 2 t2
E1.10 Fourier Series and Transforms (2014-5559)
2
1+4π 2 f 2
⇒
U (f ) =
⇒
V (f ) = e−|−f |
[from earlier]
Fourier Transform: 6 – 9 / 12
Duality
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Dual transform:
Suppose v(t) = U (t), then
R∞
V (f ) = t=−∞ v(t)e−i2πf t dτ
R∞
V (g) = t=−∞ U (t)e−i2πgt dt [substitute f = g , v(t) = U (t)]
R∞
= f =−∞ U (f )e−i2πgf df
[substitute t = f ]
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
So:
= u(−g)
v(t) = U (t) ⇒ V (f ) = u(−f )
Example:
u(t) = e−|t|
v(t) =
2
1+4π 2 t2
E1.10 Fourier Series and Transforms (2014-5559)
2
1+4π 2 f 2
⇒
U (f ) =
⇒
V (f ) = e−|−f | = e−|f |
[from earlier]
Fourier Transform: 6 – 9 / 12
Time Shifting and Scaling
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 10 / 12
Time Shifting and Scaling
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Time Shifting and Scaling:
Suppose v(t) = u(at + b)
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 10 / 12
Time Shifting and Scaling
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Time Shifting and Scaling:
Suppose v(t) = u(at + b), then
V (f ) =
E1.10 Fourier Series and Transforms (2014-5559)
R∞
t=−∞
u(at + b)e−i2πf t dt
Fourier Transform: 6 – 10 / 12
Time Shifting and Scaling
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Time Shifting and Scaling:
Suppose v(t) = u(at + b), then
V (f ) =
E1.10 Fourier Series and Transforms (2014-5559)
R∞
t=−∞
u(at + b)e−i2πf t dt
[now sub τ = at + b]
Fourier Transform: 6 – 10 / 12
Time Shifting and Scaling
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Time Shifting and Scaling:
Suppose v(t) = u(at + b), then
R∞
[now sub τ = at + b]
u(at + b)e−i2πf t dt
R∞
−i2πf ( τ −b
a ) 1 dτ
= sgn(a) τ =−∞ u(τ )e
a
V (f ) =
E1.10 Fourier Series and Transforms (2014-5559)
t=−∞
Fourier Transform: 6 – 10 / 12
Time Shifting and Scaling
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Time Shifting and Scaling:
Suppose v(t) = u(at + b), then
R∞
[now sub τ = at + b]
u(at + b)e−i2πf t dt
R∞
−i2πf ( τ −b
a ) 1 dτ
= sgn(a) τ =−∞ u(τ )e
a
(
1
a>0
note that ±∞ limits swap if a < 0 hence sgn(a) =
−1 a < 0
V (f ) =
E1.10 Fourier Series and Transforms (2014-5559)
t=−∞
Fourier Transform: 6 – 10 / 12
Time Shifting and Scaling
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Time Shifting and Scaling:
Suppose v(t) = u(at + b), then
R∞
[now sub τ = at + b]
u(at + b)e−i2πf t dt
R∞
−i2πf ( τ −b
a ) 1 dτ
= sgn(a) τ =−∞ u(τ )e
a
(
1
a>0
note that ±∞ limits swap if a < 0 hence sgn(a) =
−1 a < 0
b R∞
1 i 2πf
−i2π fa τ
a
u(τ
)e
= |a| e
dτ
τ =−∞
V (f ) =
E1.10 Fourier Series and Transforms (2014-5559)
t=−∞
Fourier Transform: 6 – 10 / 12
Time Shifting and Scaling
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Time Shifting and Scaling:
Suppose v(t) = u(at + b), then
R∞
[now sub τ = at + b]
u(at + b)e−i2πf t dt
R∞
−i2πf ( τ −b
a ) 1 dτ
= sgn(a) τ =−∞ u(τ )e
a
(
1
a>0
note that ±∞ limits swap if a < 0 hence sgn(a) =
−1 a < 0
b R∞
1 i 2πf
−i2π fa τ
a
u(τ
)e
= |a| e
dτ
τ =−∞
b
1 i 2πf
= |a| e a U fa
V (f ) =
E1.10 Fourier Series and Transforms (2014-5559)
t=−∞
Fourier Transform: 6 – 10 / 12
Time Shifting and Scaling
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
Fourier Transform: u(t) = −∞ U (f )ei2πf t df
R∞
U (f ) = t=−∞ u(t)e−i2πf t dt
[Fourier Synthesis]
[Fourier Analysis]
Time Shifting and Scaling:
Suppose v(t) = u(at + b), then
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
[now sub τ = at + b]
u(at + b)e−i2πf t dt
R∞
−i2πf ( τ −b
a ) 1 dτ
= sgn(a) τ =−∞ u(τ )e
a
(
1
a>0
note that ±∞ limits swap if a < 0 hence sgn(a) =
−1 a < 0
b R∞
1 i 2πf
−i2π fa τ
a
u(τ
)e
= |a| e
dτ
τ =−∞
b
1 i 2πf
= |a| e a U fa
V (f ) =
So:
t=−∞
v(t) = u(at + b)
E1.10 Fourier Series and Transforms (2014-5559)
⇒ V (f ) =
b
1 i 2πf
a U
e
|a|
f
a
Fourier Transform: 6 – 10 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t)
0.4
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t)
0.4
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
U (f ) =
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
−i2πf t
u(t)e
dt
−∞
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t)
0.4
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
U (f ) =
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
R∞
−∞
−i2πf t
u(t)e
dt =
√ 1
2πσ 2
R∞
−∞
2
t
− 2σ
2 −i2πf t
e
e
dt
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
u(t)
0.4
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
• Dirac Delta Function
• Dirac Delta Function:
U (f ) =
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
=
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
u(t)e
R∞
−∞
√ 1
2πσ 2
−∞
0.4
u(t)
2
R ∞ − t −i2πf t
1
√
dt = 2πσ2 −∞ e 2σ2 e
dt
− 2σ1 2 (t2 +i4πσ 2 f t)
−i2πf t
e
dt
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
• Dirac Delta Function
• Dirac Delta Function:
U (f ) =
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
=
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
=
R∞
u(t)e
R∞
−∞
√ 1
2πσ 2
√ 1
2πσ 2
−∞
R∞
−∞
0.4
u(t)
2
R ∞ − t −i2πf t
1
√
dt = 2πσ2 −∞ e 2σ2 e
dt
− 2σ1 2 (t2 +i4πσ 2 f t)
−i2πf t
e
dt
2
2
2
2
2
2
1
− 2σ2 t +i4πσ f t+(i2πσ f ) −(i2πσ f )
e
dt
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
• Dirac Delta Function
• Dirac Delta Function:
U (f ) =
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
=
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
=
=
R∞
u(t)e
R∞
−∞
√ 1
2πσ 2
−∞
e
dt
2
2
2
2
2
2
1
− 2σ2 t +i4πσ f t+(i2πσ f ) −(i2πσ f )
R∞
√ 1
e
2πσ 2 −∞
2
2
1
i2πσ
f
(
)
√ 1
e 2σ2
2πσ 2
0.4
u(t)
2
R ∞ − t −i2πf t
1
√
dt = 2πσ2 −∞ e 2σ2 e
dt
− 2σ1 2 (t2 +i4πσ 2 f t)
−i2πf t
R∞
−∞
− 2σ1 2 (t+i2πσ 2 f )
e
2
dt
dt
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
• Dirac Delta Function
• Dirac Delta Function:
U (f ) =
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
=
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
u(t)e
R∞
−∞
√ 1
2πσ 2
−∞
e
dt
2
2
2
2
2
2
1
− 2σ2 t +i4πσ f t+(i2πσ f ) −(i2πσ f )
R∞
1
√
= 2πσ2 −∞ e
2
2
1
i2πσ
f
(
)
√ 1
= e 2σ2
2πσ 2
2
2
1
(i)
i2πσ
f
(
)
2
2σ
=e
0.4
u(t)
2
R ∞ − t −i2πf t
1
√
dt = 2πσ2 −∞ e 2σ2 e
dt
− 2σ1 2 (t2 +i4πσ 2 f t)
−i2πf t
R∞
−∞
− 2σ1 2 (t+i2πσ 2 f )
e
2
dt
dt
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
• Dirac Delta Function
• Dirac Delta Function:
U (f ) =
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
=
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
u(t)e
R∞
−∞
√ 1
2πσ 2
−∞
e
dt
2
2
2
2
2
2
1
− 2σ2 t +i4πσ f t+(i2πσ f ) −(i2πσ f )
R∞
1
√
= 2πσ2 −∞ e
2
2
1
i2πσ
f
(
)
√ 1
= e 2σ2
2πσ 2
2
2
1
(i)
i2πσ
f
(
)
2
2σ
u(t)
R∞
−∞
− 2σ1 2 (t+i2πσ 2 f )
e
2
dt
dt
=e
[(i) uses a result from complex analysis theory that:
R ∞ − 1 (t+i2πσ2 f )2
R ∞ − 1 t2
1
1
2
√
e 2σ
e 2σ2 dt = 1]
dt = √
2πσ 2
0.4
2
R ∞ − t −i2πf t
1
√
dt = 2πσ2 −∞ e 2σ2 e
dt
− 2σ1 2 (t2 +i4πσ 2 f t)
−i2πf t
−∞
2πσ 2
−∞
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
• Dirac Delta Function
• Dirac Delta Function:
U (f ) =
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
=
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
u(t)e
R∞
−∞
√ 1
2πσ 2
2
R ∞ − t −i2πf t
1
√
dt = 2πσ2 −∞ e 2σ2 e
dt
− 2σ1 2 (t2 +i4πσ 2 f t)
−i2πf t
−∞
e
dt
2
2
2
2
2
2
1
− 2σ2 t +i4πσ f t+(i2πσ f ) −(i2πσ f )
R∞
1
√
= 2πσ2 −∞ e
2
R ∞ − 1 (t+i2πσ2 f )2
2
1
i2πσ
f
1
(
)
√
e 2σ2
dt
= e 2σ2
2πσ 2 −∞
2
2
1
(i)
i2πσ
f
(
)
− 12 (2πσf )2
2
2σ
=e
=e
[(i) uses a result from complex analysis theory that:
R ∞ − 1 (t+i2πσ2 f )2
R ∞ − 1 t2
1
1
2
√
e 2σ
e 2σ2 dt = 1]
dt = √
2πσ 2
u(t)
0.4
dt
−∞
2πσ 2
−∞
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
• Dirac Delta Function
• Dirac Delta Function:
U (f ) =
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
=
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
u(t)e
R∞
−∞
√ 1
2πσ 2
2
R ∞ − t −i2πf t
1
√
dt = 2πσ2 −∞ e 2σ2 e
dt
− 2σ1 2 (t2 +i4πσ 2 f t)
−i2πf t
−∞
e
dt
2
2
2
2
2
2
1
− 2σ2 t +i4πσ f t+(i2πσ f ) −(i2πσ f )
R∞
1
√
= 2πσ2 −∞ e
2
R ∞ − 1 (t+i2πσ2 f )2
2
1
i2πσ
f
1
(
)
√
e 2σ2
dt
= e 2σ2
2πσ 2 −∞
2
2
1
(i)
i2πσ
f
(
)
− 12 (2πσf )2
2
2σ
=e
=e
[(i) uses a result from complex analysis theory that:
R ∞ − 1 (t+i2πσ2 f )2
R ∞ − 1 t2
1
1
2
√
e 2σ
e 2σ2 dt = 1]
dt = √
2πσ 2
−∞
2πσ 2
σ=1
0.2
0
-4
-2
E1.10 Fourier Series and Transforms (2014-5559)
0
Time (s)
2
4
−∞
1/(2πσ)=0.159
1
|U(f)|
u(t)
0.4
dt
0.5
0
-0.6
-0.4
-0.2
0
0.2
Frequency (Hz)
0.4
0.6
Fourier Transform: 6 – 11 / 12
Gaussian Pulse
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
Gaussian Pulse: u(t) =
2
t
− 2σ
2
√ 1
e
2
2πσ
R∞
This is a Normal (or Gaussian) probability distribution, so −∞ u(t)dt = 1.
• Dirac Delta Function
• Dirac Delta Function:
U (f ) =
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
=
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
R∞
u(t)e
R∞
−∞
√ 1
2πσ 2
2
R ∞ − t −i2πf t
1
√
dt = 2πσ2 −∞ e 2σ2 e
dt
− 2σ1 2 (t2 +i4πσ 2 f t)
−i2πf t
−∞
e
dt
2
2
2
2
2
2
1
− 2σ2 t +i4πσ f t+(i2πσ f ) −(i2πσ f )
R∞
1
√
= 2πσ2 −∞ e
2
R ∞ − 1 (t+i2πσ2 f )2
2
1
i2πσ
f
1
(
)
√
e 2σ2
dt
= e 2σ2
2πσ 2 −∞
2
2
1
(i)
i2πσ
f
(
)
− 12 (2πσf )2
2
2σ
=e
=e
[(i) uses a result from complex analysis theory that:
R ∞ − 1 (t+i2πσ2 f )2
R ∞ − 1 t2
1
1
2
√
e 2σ
e 2σ2 dt = 1]
dt = √
2πσ 2
−∞
2πσ 2
σ=1
0.2
0
-4
-2
0
Time (s)
2
4
−∞
1/(2πσ)=0.159
1
|U(f)|
u(t)
0.4
dt
0.5
0
-0.6
-0.4
-0.2
0
0.2
Frequency (Hz)
0.4
0.6
Uniquely, the Fourier Transform of a Gaussian pulse is a Gaussian pulse.
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 11 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 12 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
R∞
◦ Forward transform (analysis): U (f ) = t=−∞ u(t)e−i2πf t dt
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 12 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
R∞
◦ Forward transform (analysis): U (f ) = t=−∞ u(t)e−i2πf t dt
⊲ U (f ) is the spectral density function (e.g. Volts/Hz)
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 12 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
R∞
◦ Forward transform (analysis): U (f ) = t=−∞ u(t)e−i2πf t dt
⊲ U (f ) is the spectral density function (e.g. Volts/Hz)
• Dirac Delta Function:
R∞
◦ δ(t) is a zero-width infinite-height pulse with −∞ δ(t)dt = 1
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 12 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
R∞
◦ Forward transform (analysis): U (f ) = t=−∞ u(t)e−i2πf t dt
⊲ U (f ) is the spectral density function (e.g. Volts/Hz)
• Dirac Delta Function:
R∞
◦ δ(t) is a zero-width infinite-height pulse with −∞ δ(t)dt = 1
R∞
◦ Integral: −∞ f (t)δ(t − a) = f (a)
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 12 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
R∞
◦ Forward transform (analysis): U (f ) = t=−∞ u(t)e−i2πf t dt
⊲ U (f ) is the spectral density function (e.g. Volts/Hz)
• Dirac Delta Function:
R∞
◦ δ(t) is a zero-width infinite-height pulse with −∞ δ(t)dt = 1
R∞
◦ Integral: −∞ f (t)δ(t − a) = f (a)
◦ Scaling: δ(ct) =
E1.10 Fourier Series and Transforms (2014-5559)
1
|c| δ(t)
Fourier Transform: 6 – 12 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
R∞
◦ Forward transform (analysis): U (f ) = t=−∞ u(t)e−i2πf t dt
⊲ U (f ) is the spectral density function (e.g. Volts/Hz)
• Dirac Delta Function:
R∞
◦ δ(t) is a zero-width infinite-height pulse with −∞ δ(t)dt = 1
R∞
◦ Integral: −∞ f (t)δ(t − a) = f (a)
1
|c| δ(t)
P∞
u(t) = n=−∞
◦ Scaling: δ(ct) =
• Periodic Signals:
E1.10 Fourier Series and Transforms (2014-5559)
Un ei2πnF t
P∞
⇒ U (f ) = n=−∞ Un δ (f − nF )
Fourier Transform: 6 – 12 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
R∞
◦ Forward transform (analysis): U (f ) = t=−∞ u(t)e−i2πf t dt
⊲ U (f ) is the spectral density function (e.g. Volts/Hz)
• Dirac Delta Function:
R∞
◦ δ(t) is a zero-width infinite-height pulse with −∞ δ(t)dt = 1
R∞
◦ Integral: −∞ f (t)δ(t − a) = f (a)
1
|c| δ(t)
P∞
u(t) = n=−∞
◦ Scaling: δ(ct) =
• Periodic Signals:
Un ei2πnF t
P∞
⇒ U (f ) = n=−∞ Un δ (f − nF )
• Fourier Transform Properties:
◦ v(t) = U (t)
⇒ V (f ) = u(−f )
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 12 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
R∞
◦ Forward transform (analysis): U (f ) = t=−∞ u(t)e−i2πf t dt
⊲ U (f ) is the spectral density function (e.g. Volts/Hz)
• Dirac Delta Function:
R∞
◦ δ(t) is a zero-width infinite-height pulse with −∞ δ(t)dt = 1
R∞
◦ Integral: −∞ f (t)δ(t − a) = f (a)
1
|c| δ(t)
P∞
u(t) = n=−∞
◦ Scaling: δ(ct) =
• Periodic Signals:
Un ei2πnF t
P∞
⇒ U (f ) = n=−∞ Un δ (f − nF )
• Fourier Transform Properties:
◦ v(t) = U (t)
⇒ V (f ) = u(−f )
b
1 i 2πf
◦ v(t) = u(at + b)
⇒ V (f ) = |a| e a U fa
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 12 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
R∞
◦ Forward transform (analysis): U (f ) = t=−∞ u(t)e−i2πf t dt
⊲ U (f ) is the spectral density function (e.g. Volts/Hz)
• Dirac Delta Function:
R∞
◦ δ(t) is a zero-width infinite-height pulse with −∞ δ(t)dt = 1
R∞
◦ Integral: −∞ f (t)δ(t − a) = f (a)
1
|c| δ(t)
P∞
u(t) = n=−∞
◦ Scaling: δ(ct) =
• Periodic Signals:
Un ei2πnF t
P∞
⇒ U (f ) = n=−∞ Un δ (f − nF )
• Fourier Transform Properties:
◦ v(t) = U (t)
⇒ V (f ) = u(−f )
b
1 i 2πf
◦ v(t) = u(at + b)
⇒ V (f ) = |a| e a U fa
◦ v(t) =
E1.10 Fourier Series and Transforms (2014-5559)
2
− 21 ( σt )
√ 1
e
2πσ 2
1
2
⇒ V (f ) = e− 2 (2πσf )
Fourier Transform: 6 – 12 / 12
Summary
6: Fourier Transform
• Fourier Series as
T →∞
• Fourier Transform
• Fourier Transform
Examples
• Dirac Delta Function
• Dirac Delta Function:
Scaling and Translation
• Dirac Delta Function:
Products and Integrals
• Periodic Signals
• Duality
• Time Shifting and Scaling
• Gaussian Pulse
• Summary
• Fourier Transform:
R∞
◦ Inverse transform (synthesis): u(t) = −∞ U (f )ei2πf t df
R∞
◦ Forward transform (analysis): U (f ) = t=−∞ u(t)e−i2πf t dt
⊲ U (f ) is the spectral density function (e.g. Volts/Hz)
• Dirac Delta Function:
R∞
◦ δ(t) is a zero-width infinite-height pulse with −∞ δ(t)dt = 1
R∞
◦ Integral: −∞ f (t)δ(t − a) = f (a)
1
|c| δ(t)
P∞
u(t) = n=−∞
◦ Scaling: δ(ct) =
• Periodic Signals:
Un ei2πnF t
P∞
⇒ U (f ) = n=−∞ Un δ (f − nF )
• Fourier Transform Properties:
◦ v(t) = U (t)
⇒ V (f ) = u(−f )
b
1 i 2πf
◦ v(t) = u(at + b)
⇒ V (f ) = |a| e a U fa
◦ v(t) =
2
− 21 ( σt )
√ 1
e
2πσ 2
1
2
⇒ V (f ) = e− 2 (2πσf )
For further details see RHB Chapter 13.1 (uses ω instead of f )
E1.10 Fourier Series and Transforms (2014-5559)
Fourier Transform: 6 – 12 / 12